True or False. Justify your answer with a proof or a counterexample. Assume all functions [latex]f[/latex] and [latex]g[/latex] are continuous over their domains.

1. If [latex]f(x) < 0,{f}^{\prime }(x) < 0[/latex] for all [latex]x,[/latex] then the right-hand rule underestimates the integral [latex]{\int }_{a}^{b}f(x).[/latex] Use a graph to justify your answer.

Solution

False

2.  [latex]{\int }_{a}^{b}f{(x)}^{2}dx={\int }_{a}^{b}f(x)dx{\int }_{a}^{b}f(x)dx[/latex]

3.  If [latex]f(x)\le g(x)[/latex] for all [latex]x\in \left[a,b\right],[/latex] then [latex]{\int }_{a}^{b}f(x)\le {\int }_{a}^{b}g(x).[/latex]

Solution

True

4.  All continuous functions have an antiderivative.

Evaluate the Riemann sums [latex]{L}_{4}\text{ and }{R}_{4}[/latex] for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5.  [latex]y=3{x}^{2}-2x+1[/latex] over [latex]\left[-1,1\right][/latex]

Solution

[latex]{L}_{4}=5.25,{R}_{4}=3.25,[/latex] exact answer: 4

6.  [latex]y=\text{ln}({x}^{2}+1)[/latex] over [latex]\left[0,e\right][/latex]

7.  [latex]y={x}^{2} \sin x[/latex] over [latex]\left[0,\pi \right][/latex]

Solution

[latex]{L}_{4}=5.364,{R}_{4}=5.364,[/latex] exact answer: 5.870

8.  [latex]y=\sqrt{x}+\frac{1}{x}[/latex] over [latex]\left[1,4\right][/latex]

Evaluate the following integrals.

9.  [latex]{\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx[/latex]

Solution

[latex]-\frac{4}{3}[/latex]

10.  [latex]{\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt[/latex]

11.  [latex]{\int }_{\pi \text{/}3}^{\pi \text{/}2}2 \sec (2\theta ) \tan (2\theta )d\theta[/latex]

Solution

1

12.  [latex]{\int }_{0}^{\pi \text{/}4}{e}^{{ \cos }^{2}x} \sin x \cos dx[/latex]

Find the antiderivative.

13.  [latex]\int \frac{dx}{{(x+4)}^{3}}[/latex]

Solution

[latex]-\frac{1}{2{(x+4)}^{2}}+C[/latex]

14.  [latex]\int x\text{ln}({x}^{2})dx[/latex]

15.  [latex]\int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx[/latex]

Solution

[latex]\frac{4}{3}\phantom{\rule{0.05em}{0ex}}{ \sin }^{-1}({x}^{3})+C[/latex]

16.  [latex]\int \frac{{e}^{2x}}{1+{e}^{4x}}dx[/latex]

Find the derivative.

17.  [latex]\frac{d}{dt}{\int }_{0}^{t}\frac{ \sin x}{\sqrt{1+{x}^{2}}}dx[/latex]

Solution

[latex]\frac{ \sin t}{\sqrt{1+{t}^{2}}}[/latex]

18.  [latex]\frac{d}{dx}{\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt[/latex]

19.  [latex]\frac{d}{dx}{\int }_{1}^{\text{ln}(x)}(4t+{e}^{t})dt[/latex]

Solution

[latex]4\frac{\text{ln}x}{x}+1[/latex]

20.  [latex]\frac{d}{dx}{\int }_{0}^{ \cos x}{e}^{{t}^{2}}dt[/latex]

The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($)
1980 0
1985 -5,468,750
1990 755,495
1995 -73,005
2000 -29,768
2005 -918
2010 -177

21.  If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

Solution

$6,328,113

22.  The average cost per gigabyte of RAM can be approximated by the function [latex]C(t)=8,500,000{(0.65)}^{t},[/latex] where [latex]t[/latex] is measured in years since 1980, and [latex]C[/latex] is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

23.  Find the average cost of 1GB RAM for 2005 to 2010.

Solution

$73.36

24.  The velocity of a bullet from a rifle can be approximated by [latex]v(t)=6400{t}^{2}-6505t+2686,[/latex] where [latex]t[/latex] is seconds after the shot and [latex]v[/latex] is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: [latex]0\le t\le 0.5.[/latex] What is the total distance the bullet travels in 0.5 sec?

25.  What is the average velocity of the bullet for the first half-second?

Solution

[latex]\frac{19117}{12}\text{ft/sec},\text{or}1593\text{ft/sec}[/latex]

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